Volume 28 Number 1. Keywords: emission, Eskom, log logistic distribution, goodness of fit, sulphur dioxide, Burr-type distribution

Transcription

1 Volume 28 Number 1 Quantifying South Africa s sulphur dioxide emission efficiency in coal-powered electricity generation by fitting the three-parameter log-logistic distribution Maseapei Elizabeth Girmay*, Delson Chikobvu Department of Mathematical Statistics and Actuarial Science, University of the Free State, PO Box 339, Bloemfontein 9300, South Africa Abstract This paper fits the three-parameter log-logistic () distribution to sulphur dioxide (SO 2) monthly emissions in kilograms per gigawatt hour (kg/gwh) and in milligrams per cubic nano metre (mg/nm 3 ), at 13 of Eskom s coal fired power-generating stations in South Africa. The aim is to quantify and describe the emission of sulphur dioxide at these stations using a statistical distribution, and to also estimate the probabilities of extreme emissions and exceedances (emissions above a certain threshold). Using the distribution is proposed as such a distribution. The log-logistic distribution is a special form of a Burrtype distribution. Various goodness-of-fit measures, including the Kolmogorov Smirnov, the Anderson Darling and some graphical tests, are employed to test if the distribution is a good fit to the data. The maximum likelihood method is used to estimate the parameters. The distribution fit is important as it then becomes possible to quantify and manage the SO 2 emissions effectively. The distribution, which is compared with three other distributions, gave the best overall fit to most of the power stations. Keywords: emission, Eskom, log logistic distribution, goodness of fit, sulphur dioxide, Burr-type distribution Highlights Quantification of SO 2 emissions in terms of a statistical distribution Calculating the probability of SO 2 emissions exceeding certain specified limits Ranking power stations in terms of SO 2 emissions efficiency Journal of Energy in Southern Africa 28(1): DOI: * Corresponding author: Tel: +27 (0) girmayme@ufs.ac.za 91

2 1. Introduction Eskom is South Africa s electricity public utility, established in 1923 as the Electricity Supply Commission by the government of South Africa. It is the largest producer of electricity in Africa, and among the top seven utilities in the world in terms of generation capacity and among the top nine in terms of sales. Eskom generates approximately 95% of electricity used in South Africa. About 95% of its generating capacity comes from coal. Ash emissions from Eskom s coal-fired power stations have been reduced by more than 90% since the early 1980s due to the installation of efficient pollution abatement technology and the decommissioning of older plants (Eskom Emission Monitoring, 2012). At present, Eskom has 13 coal-fired generating stations giving various emissions, including sulphur dioxide (SO 2). Medupi and Kusile are two new stations and are not included in the analysis. Coal-fired power stations release harmful chemicals (stack emissions) into the atmosphere, causing environmental problems. Sulphur dioxide, for example, is a precursor to acid deposition (including acid rain) and secondary particulate matter formation, and is also toxic. Eskom must comply with legally prescribed limits on a number of emissions to avoid heavy penalties. Exceeding the emission limits may result in the forced shutdown of generating units. Emission levels must therefore be monitored and alarmed continuously. A stack test is a procedure for sampling a gas stream from a single sampling location at a facility, unit, or pollution control equipment. Each of Eskom s stations could be a unit, and there could also be more than one unit at a station. The stack is used to determine a pollutant emission rate, concentration, or parameter while the facility, unit, or pollution control equipment is operating at conditions that result in the measurement of the highest emission or parameter values (prior to any control device) or at other operating conditions approved by the regulatory authority. Stack emission control programmes are effective only if emissions are controlled at the source, which requires a highly accurate monitoring schedule. Stack emissions are measured for various reasons, including to determine if emission permits requirements are met or exceeded, for emissions permit renewal, or for process control purposes. Reporting on environmental performance has several benefits, including providing management with information to help exploit the cost savings that good environmental performance usually brings; it also gives Eskom the chance to set out what they believe is significant in their environmental performance. Companies that measure (to quantify), manage and communicate their environmental performance are inherently well placed. They understand how to improve their processes, reduce their costs, comply with regulatory requirements and stakeholder expectations, and take advantage of new technologies on the market. This paper describes SO 2 emissions at 13 Eskom coalfired power stations measured by fitting a three-parameter log-logistic () distribution. The maximum likelihood method is used to estimate the parameters. The is compared with three other distributions, namely the normal, log-normal and three-parameter Weibull. A literature review follows in Section 2; methodology is outlined in Section 3; the data and results are given in Section 4 and conclusions in Section Literature review Little if any modelling of pollutant emissions in South Africa is done. Geogopolous et al. (1982) stated that the answer to the question of which distribution best fits the air quality/emissions data was shown to depend in general on the pollutant, the time period of interest, the averaging time of the data, the location and other factors. Generally there is no priori reason to choose one particular distribution over the other (Seinfeld et al., 1998). Yi Zhang et al. (1994) investigated the statistical distribution of on-road carbon monoxide and hydrocarbon emissions from various locations in the United States, and found that the emissions are statistically gamma distributed. Rumburg et al. (2001) investigated the statistical distribution of particulate matter in Spokane, Washington and concluded that the distribution was best fitted by a three-parameter lognormal distribution and a generalised extreme value distribution. Hadley et al. (2003) investigated the distribution of annual mean daily SO 2 in the United Kingdom and found the log-normal distribution to be a better fit for the data than the normal. One of the most important papers on modelling greenhouse gas emissions is that of Smith (1989), where he applies the extreme value theory to the study of hourly readings of ozone in Houston, Texas, since excessive levels of ozone are taken to indicate high air pollution. Smith concluded that the exponential distribution gave a poor fit in the upper tail whereas the generalised Pareto distribution seemed adequate. The method of estimation of parameters for the chosen statistical distribution is also important when emission data is available. Ashkar et al. (2003) compared the maximum likelihood (ML) and the generalised probability weighted moments (GPWM) in estimating the parameters of a log-logistic, and found that the ML outperformed the GPWM method over all parameter space and sample sizes. Ashkar et al. (2006) compared the method of generalised moments, ML, the methods of the GPWM and the method of log moments for the estimation of the parameters and quantiles in the two-parameter log-logistic. Their simulation results showed that the GM method outperformed the other competitive methods in the two-parameter log-logistic case when the moment orders are appropriately chosen. Abbas et 92 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

3 al. (2015) proposed the Bayesian method using the metropolis algorithm within the Gibbs sampling under the reference prior to estimate the parameters of the log-logistic. Singh et al. (1993) developed a new competitive method of estimating parameters of the log-logistic based on the principle of maximum entropy (POME) using the Monte Carlo simulated data, and compared it to the methods of moments (MOM), ML, and the probability weighted moments (PWM). They concluded that POME yielded the least parameter bias for all sample sizes. Other parameter estimation problems for the log-logistic distribution are addressed by, among others, Kantam and Srinivasa (2002), Balakrishnan et al. (1987), and Tiku and Suresh (1992). The log-logistic distribution is a special type of a Burr-type XII distribution. Burr (1942) introduced twelve different forms of cumulative distribution functions for modelling data, of which Burr-type X and Burr-type XII received the maximum attention. There is also a thorough analysis of Burr-type XII distribution in Rodriguez (1977), and see also Wingo (1993). Burr-type distributions are very flexible and can be adapted to fit many situations. In this paper the three-parameter log-logistic distribution from the Burr-type XII family is used. The normal is one of the most commonly used distributions where there is a large data set. Log-normal is one of the distributions commonly mentioned in the literature on emissions; the variance of the log-normal increases with the mean. The three-parameter Weibull distribution is also one of the commonly used reported distributions, and it is also heavy-tailed. 3. Research methodology This section describes the statistical distribution used to describe the SO 2 emissions data and the method of estimating the parameters. The log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is the probability distribution of a random variable whose logarithm has a logistic distribution. It is generated by a transformation of logistic variable, just like the log-normal distribution is obtained from the normal distribution. It is similar in shape to the log-normal distribution but has heavier tails. The log-logistic distribution is a special case of the Burr-type XII distribution (Burr, 1942) and also a special case of the Kappa distribution (Mielke & Johnson, 1973). The Burr distribution is very relevant in the study of atmospheric emissions, it is more flexible and has heavier tails and is then able to model extreme emissions. Its cumulative distribution function can be written in closed form, unlike that of the log-normal. The distribution has a probability density function given as in Equation 1. 2 αα ff(xx) = αα xx γγ αα xx γγ αα > 0, > 0, γγ xx < + (1) and a cumulative distribution given by Equation 2: FF(xx) = 1 + xx γγ αα 1 αα > 0, > 0, γγ xx < + (2) where > 0 is a scale parameter, αα > 0 is a shape parameter and γγ RR is a location parameter. The distribution is unimodal when the shape parameter αα > 1 and its dispersion decreases as the shape parameter αα increases. The mean and variance of the distribution are given by Equations 3 and 4 respectively: EE(XX) = γγ , 1 1 (3) αα αα VVVVVV(xx) = 2 BB 1 + 2, 1 2 αα αα BB , 1 1 αα αα (4) 3.1 Parameter estimation The parameters are estimated using the ML method. The ML is asymptotically normal, has the smallest asymptotic variance, and is asymptotically efficient and optimal. With the ML approach, the distributions of the estimators become more and more concentrated near the true value of the parameter being estimated as the sample size (n) increases. Maximum likelihood method: The probability density function of the distribution is given in Equation (1) above, with the log likelihood given in Equation 5: LL(xx; αα,, γγ) = nnnnnnnn nnnnnnnn + nn (αα 1) llll xx ii γγ ii=1 nn 2 llll 1 + xx ii γγ ii=1 αα (5) where nn is the sample size. Taking the partial derivatives of each parameter gives Equations 6 8: (xx;αα,,γγ) 2 = nn + nn llll αα xx ii γγ ii=1 xx ii γγ αα llll xx ii γγ nn ii=1 (6) 1+ xx ii γγ αα (xx;αα,,γγ) (xx;αα,,γγ) = nn αα 1 + 2αα nn xx ii γγ αα ii=1 (7) 1+ xx ii γγ αα nn 1 = (αα 1) ii=1 + 2αα nn ii=1 xx ii 1 xx ii γγ αα 1 1+ xx ii γγ αα (8) 93 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

4 The maximum likelihood estimates αα, and γγ are obtained by setting each of the above equations to zero and solving them simultaneously. Optimisation computer algorithms such as the Newton- Raphson, Nelder-Mead, and simulated annealing are often used to arrive at the estimates. The normal, log-normal and three-parameter Weibull distribution parameters are estimated in a similar way Goodness of fit tests The Anderson-Darling, Chi Squared, and Kolmogorov-Smirnov tests are used to test if the is a good distribution to fit the data. The probability density function (PDF) plots, cumulative distribution function (CDF) plots, together with the probability-probability (P-P) and quantile-quantile (Q-Q) plots are also used as tests. The Kolmogorov-Smirnov test is a nonparametric test for the equality of continuous, one-dimensional probability or statistical distributions that can be used to compare a sample of observations with a reference probability distribution or to compare two samples. It tries to determine if two data sets differ significantly, and has the advantage of making no assumption about the statistical distribution of data. It quantifies a distance between the empirical distribution function of the sample and the theoretical cumulative distribution function, or between the empirical distribution functions of two samples. The test statistic of the Kolmogorov Smirnov is given by Equation 9: DD nn = mmmmmm mmmmmm ii=1.nn 1 nn FF (xx ii ) ; mmmmmm ii=1.nn FF (xx ii ) ii 1 nn (9) which is the largest vertical difference between the theoretical and empirical CDF for all values of xx (Evans et al., 2008), where nn is the sample size. The Anderson Darling test is also a statistical test of whether a given sample of data is drawn from a given probability distribution. It is a modification of the Kolmogorov-Smirnov test and gives more weight to the tails of a statistical distribution than that test. In its basic form, the test does not assume a particular parametric form of the distribution being tested, in which case the test and its set of critical values is distribution free. The test is, however, most often used in context where a family of distributions is being tested, in which case the parameters of that family need to be estimated and account must be taken of this in adjusting either the test-statistic or its critical values. When applied to testing if a normal distribution adequately describes a set of data, it is one of the most powerful statistical tools for detecting most departures from normality. The test assesses whether a sample comes from a specified distribution. It compares the observed CDF with the expected CDF and is defined in Equation 10: AA 2 nn = nn FF nn (xx) FF (xx) 2 ddff (xx) FF (xx)[1 FF (xx)] (10) with the computational formula given by Equation 11: nn AA 2 nn = 2ii 1 llll FF (xx nn ii ) + llll 1 FF (xx nn+1 ) nn ii=1 (11) where nn is the sample size. According to Abbas et al. (2012), the Anderson-Darling test is superior where there is greater concern for the extreme values in the data. This is very relevant when considering atmospheric emissions data. The P-P and Q-Q plots are graphical methods used to test the fit of the distributions to the data. The P-P plot assesses whether or not the data set follows the specified distribution. The P-P plot compares the CDF of the distributions by plotting the theoretical values and the points of the empirical distribution against it. The Q-Q plot compares the distributions by plotting their quantiles against each other. With the Q-Q plot, the data are plotted against the theoretical distribution. For both the P-P and Q-Q plots, if the empirical distribution is close to the theoretical distribution, the graph will be a straight line (Beirlant et al., 2004). Departures from the straight line indicate the departures from the theoretical distribution. 4. Results and discussions The data is from Eskom, for the period January 2005 to January 2012.The efficiency of the power station cannot be determined by absolute emissions, but by the amount of SO 2 emitted in kilograms per gigawatt hours (kg/gwh) of electricity sent out (relative emissions). To accommodate the non-stationarity in the data, the total emissions of SO 2 emitted (in kilograms or milligrams) per month per power station is divided by the total amount of units of power sent out per power station per month or by the volume. The derived data for the power stations, therefore, represents the amount of emission emitted in kilograms to send out one unit of power in gigawatt hours or in milligrams per cubic nano metre (mg/nm 3 ). The resultant variables are a measure of efficiency of the station in emitting SO 2 and will be used for the analysis. Time plots are useful for checking obvious patterns in the data. The plots represent evolving efficiency at the power stations. Time plots of Lethabo and Camden power stations SO 2 kg/gwh and SO 2 mg/nm 3 data are given in Figures 1 and 2 respectively. The rest of the time plots are given in supplementary information. 94 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

5 SO 2 /GWh Sep-05 May-06 Jan-07 Sep-07 May-08 Jan-09 Sep-09 May-10 Jan-11 (a) Sep-11 SO 2 kg/gwh Apr-06 Dec-06 Aug-07 Apr-08 Dec-08 Aug-09 Apr-10 Dec-10 (b) Aug-11 Figure 1: Monthly SO2 in kg/gwh emissions, where (a) = Lethabo and (b) = Camden. SO 2 mg/nm (a) Jan-05 Aug-05 Mar-06 Oct-06 May-07 Dec-07 Jul-08 Feb-09 Sep-09 Apr-10 Nov-10 Jun-11 Jan-12 SO 2 mg/nm (b) Oct-06 Apr-07 Oct-07 Apr-08 Oct-08 Apr-09 Oct-09 Apr-10 Oct-10 Apr-11 Oct-11 Figure 2: Monthly SO2 in mg/nm 3 emissions, where (a) = Lethabo and (b) = Camden. Table 1: Descriptive statistics of SO2 in kg/gwh. Station Mean SO 2 (kg/gwh) Standard deviation SO 2 (kg/gwh) Skewness SO 2 (kg/gwh) Kurtosis SO 2 (kg/gwh) Arnot Camden Duhva Grootvlei Hendrina Kendal Komati Kriel Lethabo Majuba Matimba Matla Tutuka Journal of Energy in Southern Africa Vol 28 No 1 February 2017

6 Arnot, Komati, and Kriel have average monthly SO 2 emissions of around 6000 kg/gwh; Duhva, Grootvlei, Hendrina, Hendrina, Lethabo, Majuba, Matimba, Matla and Tutuka average 8000 kg/gwh; and Camden, Kandal and Matimba average emission around 9000 kg/gwh. For SO 2 in mg/nm 3 most power stations (Camden, Duhva, Hendrina, Kendal, Lethabo, Majuba, Matla and Tutuka) have an average monthly emission of 2000, and Arnot, Grootvlei, Komati and Kriel Graphs for Lethabo and Camden are given in Figure 2, the remainder in the supplementary information. Matimba had the highest monthly average of 3000 mg/nm 3. Matimba received the 2011 National Association of Clean Air award for consistent reduction of point source particulate emission (COPFIT Fact Sheet, Eskom, 2012), but it seems not to be the case for SO 2 emissions. An augmented Dickey-Fuller test for stationarity was carried out for all of the stations for both SO 2 in kg/gwh and SO 2 in mg/nm 3 at 5% significance level, showing that all stations were stationary, that none of the stations had a trend. Table 1 gives the descriptive statistics of the power stations for SO 2 in kg/gwh, and Table 2 gives the statistics for SO 2 in mg/nm 3. Figures 3 and 4 give graphical plots of the monthly means for each station for both SO 2 in kg/gwh and SO 2 in mg/nm 3 from the worst to the better emitters. Station Mean SO 2 (mg/nm 3 ) Table 2: Descriptive statistics of SO2 in mg/nm 3. Standard deviation SO 2 (mg/nm 3 ) Skewness SO 2 (mg/nm 3 ) Kurtosis SO 2 (mg/nm 3 ) Arnot Camden Duhva Grootvlei Hendrina Kendal Komati Kriel Lethabo Majuba Matimba Matla Tutuka Figure 3: SO2 in kg/gwh monthly means in an ascending order of efficiency. 96 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

7 Figure 4: The SO2 in mg/nm 3 monthly means in an ascending order of efficiency. Table 3: Abatement technology used in Eskom power stations (Eskom, 2012). Power station Arnot Camden Duhva Unit 1 3 Duhva Unit 4 6 Grootvlei Units 1, 5, 6 Grootvlei Units 2, 3, 4 Hendrina Kendal Komati Kriel Lethabo Majuba s Matimba Matla Tutuka Abatement technology Fabric filter plants Fabric filter plants Fabric filter plants Fabric filter plants Fabric filter plants Fabric filter plant Electrostatic precipitators From Figures 3 and 4 it can be concluded that Matimba and Kendal are the least efficient stations in emitting SO 2, with Arnot and Kriel the most efficient. Eskom installed abatement technologies at each power station to reduce the ash emissions. These technologies are said to have an efficiency of at 99%, and over 99.9% in many cases (COPFIT Fact Sheet, Eskom, 2012). These abatement technologies are given in Table 3. From Figures 3 and 4 in conjunction with Table 3, it can be concluded that the stations that use an electrostatic precipitators and flue gas conditioning technology are less efficient than the ones using fabric filter plants, but other factors also affect emission efficiency, including the age of the plant and the quality of coal used. To further describe the distribution of the data we look at skewness and kurtosis of the data. The skewness is defined by Equation 12: ss = EE(xx μμ)3 (12) σσ 3 where μ and σσ are the mean and standard deviation of emission efficiency variable xx. The skewness shows how symmetric the data is around the mean. A symmetric data has skewness near zero, while negative skewness indicates that the data is spread more to the left of the mean and positive skewness indicates that the data is spread to the right of the mean. By negative-skewed, the left tail is long relative to the right tail, and by positive-skewed the right tail is long relative to the left. For SO 2 in kg/gwh, Camden, Grootvlei Kendal, Komati and Tutuka 97 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

8 have a negative skewness, indicating that their data is more spread to the left of the mean. For the rest of the stations the data is more spread to the right of the mean. For SO 2 in mg/nm 3, Arnot, Grootvlei, and Matimba have a negative skew, and all the other stations have a positive skew. Only Grootvlei is confirmed with a negative skewness in both data sets. The kurtosis is defined by Equation 13: kk = EE(xx μμ)4 σσ 4 (13) where μ and σσ are the mean and standard deviation of xx. Kurtosis is a measure of whether the data sets are heavy- or light-tailed relative to the normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or a lack of outliers. If a distribution has a kurtosis less than three, its tail is shorter and thinner and its peak is flatter and broader than the normal distribution (Brown 2011). For SO 2 in kg/gwh, only Kendal, Lethabo and Matla have a kurtosis higher than three, indicating that they have a central peak higher and sharper and that their tails are longer and fatter than that of normal distribution. All the other stations have a tail shorter and thinner and their peaks are flatter and broader than the normal distribution. Since most stations are positively skewed for both SO 2 in kg/gwh and SO 2 in mg/nm 3, a rightskewed distribution needs to be considered, indicating that it could be reasonable to fit a distribution to the data. Tables 4 and 5 give the parameter estimates of the distribution for both the SO 2 in kg/gwh and SO 2 in mg/nm 3. The parameters are estimated using the ML estimation method. Table 4: Parameter estimates of SO2 kg/gwh using distribution. Station SO 2 kg/gwh Arnot αα = = γγ = Camden αα = 1.47EE + 8 = 1.31EE + 11 γγ = 1.31EE + 11 Duhva αα = = γγ = Grootvlei αα = 5.54EE + 7 = 5.44EE + 10 γγ = 5.44EE + 1 Hendrina αα = = γγ = Kendal αα = 1.30EE + 8 = 8.56EE + 10 γγ = 8.56EE + 10 Komati αα = 1.58EE + 6 = 1.63EE + 9 γγ = 1.63EE + 9 Kriel αα = 1.47EE + 8 = 1.31EE + 11 γγ = 1.31EE + 11 Lethabo αα = = γγ = Majuba αα = = γγ = Matimba αα = = γγ = Matla αα = = γγ = Tutuka αα = 6.78EE + 7 = 3.33EE + 10 γγ = 3.33EE + 10 Table 5: Parameter estimates of SO2 mg/nm 3 using distribution. Station SO 2 kg/gwh Arnot αα = 1.59EE + 8 = 1.54EE + 8 γγ = 1.54EE + 10 Camden αα = = γγ = Duhva αα = = γγ = Grootvlei αα = 2.08EE + 8 = 3.72EE + 10 γγ = 3.72EE + 10 Hendrina αα = 3.8 = γγ = Kendal αα = = γγ = Komati αα = = γγ = Kriel αα = = γγ = Lethabo αα = = γγ = Majuba αα = = γγ = Matimba αα = 3.9EE + 8 = 6.47EE + 10 γγ = 6.47EE + 10 Matla αα = = γγ = Tutuka αα = = γγ = Journal of Energy in Southern Africa Vol 28 No 1 February 2017

9 The normal, log-normal and three-parameter Weibull distributions were also fitted to the data for comparisons. Figure 4 shows the fit to the Lethabo station, with the comparisons for the distributions. Probability-probability and quantile-quantile plots: The P-P and Q-Q plots are graphical methods used to test the fit of the distributions to the data. Departures from the straight line indicates departures from the theoretical distribution. Figure 5 (split over this page and the next) gives the PDF, CDF, P-P and Q-Q plots of Lethabo station for both SO 2 kg/gwh and SO 2 mg/nm 3. Lognor Lognor- Nor Nor- Logn No 3L Lognor Nor- Lognor Nor- 99 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

10 Lognor No Nor- Lognor- Nor- Lognor Nor- Lognor- Figure 5: Distribution fit to the SO2 kg/gwh and SO2 Nm 3 for Lethabo station. The PDF, CDF together with the P-P and Q-Q plots shows a better fit for the log-logistic compared to the other distributions. Looking at the Q-Q plots it can be observed that the quantiles of and the three-parameter Weibull are closer to the straight line compared to the normal and the log-normal distributions. A goodness of fit is also done on the stations. To test for the goodness of fit, the Kolmogorov-Smirnov and Anderson-Darling tests are considered, and the results for these are given here. The Kolmogorov-Smirnov critical value at 0.05 and 0.01 level of significance for Arnot, Duhva, Hendrina, Kendal, Kriel, Lethabo, Majuba, Matimba, Matla and Tutuka are and respectively, and for Camden its and For Grootvlei and Komati, since their emissions were recorded from April 2009, the critical values are and respectively. The Anderson-Darling critical values at 0.05 and 0.01 significance level are and respectively. The null and alternative hypotheses are given as: HH 0 : The data follows the three-parameter log-logistic distribution. HH aa : The data do not follow the three-parameter log-logistic distribution. Table 6 gives the Kolmogorov-Smirnov and Anderson-Darling test results for each station. 100 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

11 Table 6: Anderson-Darling and Kolmogorov-Smirnov test results (). SO 2 kg/gwh SO 2 mg/nm 3 Statistic p-value Reject? Statistic p-value Reject? Arnot AD No No KS No No Camden AD No No KS No No Duhva AD No No KS No No Grootvlei AD No No KS No No Hendrina AD No No KS No No Kendal AD No No KS No No Komati AD No No KS No No Kriel AD No No KS No No Lethabo AD No No KS No No Majuba AD No No KS No No Matimba AD No No KS No No Matla AD No No KS No No Tutuka AD No No KS No No Both the Anderson-Darling and Kolmogorov- Smirnov tests do not reject the null hypothesis for any of the stations for both SO 2 kg/gwh and SO 2 mg/nm 3 at 5% and 1% significance level. Supplementary information gives the QQ plots for all the other stations. Tables 7 and 8 give the probabilities of exceedances above a given threshold, where tt is the threshold and PP(XX > tt) is the probability of the exceedances. Tables 7 and 8 show that Grootvlei, Kendal and Komati have a probability of almost zero for exceeding 5000 kg/gwh SO 2 emission level per month, with Arnot, Grootvlei Majuba having a probability of almost zero for exceeding 1000 mg/nm 3 SO 2 emmission level per month. Matimba, Lethabo and Matla have a probability of 1 for exceeding 7000 kg/gwh SO 2 emmission level per month with Matla, Tutuka and Duhva have a probability of 1 for exceeding 2000 mg/nm 3 SO 2 emmissions per month. This confirms that the least efficient stations with regard to emission of SO 2 are Matimba, Lethabo and Matla. Conclusions Monthly SO 2 emissions in kg/gwh and in mg/nm 3 have been considered for Eskom s 13 coal-fired power-generating stations. The fits the data of these stations best, and makes it is possible to quantify (in terms of a statistical distribution). This quantification helps to monitor and manage the SO 2 emissions effectively. The parameters of the log-logistic distribution are estimated by the ML method. Kolmogorov-Smirnov and Anderson-Darling tests are used to test for the 101 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

12 goodness of fit of the distribution for the 13 stations. The PDF, CDF, P-P and Q-Q plots are used to show how well the distribution fits the data. Considering Figures 3 and 4 together with Table 4, it can be concluded that the stations that use electrostatic precipitators and flue gas conditioning technology are less efficient than those using fabric filter plants technology, although other factors, such as the age of the plant and the quality of coal used affect emission efficiency. Table 7: The SO2 in kg/gwh probabilities of exceedances for each station (). tt = 5000 tt = 6000 tt = 7000 tt = 8000 tt = Arnot PP(XX > tt) Camden PP(XX > tt) Duhva PP(XX > tt) Grootvlei PP(XX > tt) Hendrina PP(XX > tt) Kendal PP(XX > tt) Komati PP(XX > tt) Kriel PP(XX > tt) Lethabo PP(XX > tt) Majuba PP(XX > tt) Matimba PP(XX > tt) Matla PP(XX > tt) Tutuka PP(XX > tt) Table 8: SO2 in mg/nm 3 probabilities of exceedances for each station (). tt = 1000 tt = 1500 tt = 2000 tt = 2500 tt = 3000 Arnot PP(XX > tt) Camden PP(XX > tt) Duhva PP(XX > tt) Grootvlei PP(XX > tt) Hendrina PP(XX > tt) Kendal PP(XX > tt) Komati PP(XX > tt) Kriel PP(XX > tt) Lethabo PP(XX > tt) Majuba PP(XX > tt) Matimba PP(XX > tt) Matla PP(XX > tt) Tutuka PP(XX > tt) Tables 7 and 8 show Arnot, Grootvlei and Kriel as the most efficient stations and Matimba, Lethabo and Matla as the least efficient. Looking at the results of the goodness fit, at both 5% and 1%, the null hypothesis for all stations for both the SO 2 in kg/gwh and in mg/nm 3 cannot be rejected and it is therefore concluded that the data follows the distribution. The calculated probabilities can be used to estimate costs of exceeding the given limits. The goodness of fit tests considered show that the fits the data of the Duhva, Hendrina, Kendal, Komati, Kriel, Lethabo and Matimba better than other stations. For SO 2 in mg/nm 3, the three-parameter log-logistic fits the data of Camden, Duhva, Hendrina, Komati, Kriel, and Lethabo best. From the results it shows that three-parameter log-logistic fits the positively skewed data better than the nega- 102 Journal of Energy in Southern Africa Vol 28 No 1 February 2017

13 tively skewed data. For the negatively skewed stations, it is only the three-parameter Weibull distribution that does better than the distribution. The three-parameter Weibull distribution is very close to the distribution in terms of goodness of fit. In emissions monitoring, however, concerns are more with high emissions (positively skewed), which give rise to undesirable consequences. Reporting on environmental performance has several benefits, including providing management information to help exploit the cost savings that good environmental performance usually brings and, giving Eskom the opportunity to set out what they believe is significant in their environmental performance. Further research will be done on the other Burrtype distributions to see if they will fit the data of most stations similar or better than the. The impact of the age of the plant and the quality of the coal used on atmospheric emission efficiency also requires research. Acknowledgement The authors are grateful to Eskom for providing the necessary data that made this research possible. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to Eskom. References Abbas, K.; Alamgir.; Khan, S.A.; Ali, A.; Khan, D.M.; Khalil, U Statistical analysis of wind speed data in Pakistan. World Applied Sciences Journal 18(11): Ashkar, F. and Mahdi, S Comparison of two fitting methods for log logistic. Water Resources Research 39(8): SWC 7.1 SWC 7.8 Balakrishnan, N. and Malik, H.J Best linear unbiased estimation of the location and scale parameters of the log logistic distribution. Communication in statistics A: Theory and Methods 16: Beirlant, J., Goegebeur, Y., Teugels J., De Waal, D. and Ferro C Statistics of extremes. West Sussex: John Willey & Sons. Brown, S Measures of shape: Skewness and kurtosis. Available online at: Burr, I Cumulative frequency functions. Annals of Mathematical Statistics. 13(2): Eskom Eskom emission monitoring. Available online at: Eskom Kusile and Medupi coal fired power stations under construction. Available online at: Development/ClimateChangeCOP17/Documents/Kusile_and_Medupi_coal fired_power_stations_under_construction.pdf. Eskom Climate change COP 17 fact sheet. Available online at: pany/sustainabledevelopment/climatechan- gecop17/documents/air_quality_and_cli- mate_change.pdf. Evans, D.L., Drew, J.H. and Leemis, L.M The distribution of the Kolmogorov Smirnov, Cramer von Mises and Anderson Darling tests statistics for exceptional populations with estimated parameters. Communication in Statistics Simulation and Computation 37: Georgopoulos, G.P. and Seinfeld H.J Statistical distributions of air pollutant concentrations. Environmental Science and Technology 16(7): 401A 416A. Hadley, A. and Toumi, R Assessing changes to the probability distribution of sulphur dioxide in the UK using lognormal model. Atmospheric Environment 37: Mielke, P.W., & Johnson, E.S Three parameter Kappa distribution maximum likelihood estimates and likelihood ratio tests. Monthly Weather Review 101, Mitchell, B A comparison of Chi Square and Kolmogorov Smirnov tests. Area 3(4): Available online at: ?seq=1#fndtn page_scan_tab_contents Rumburg, B., Alldredge, R. and Claiborn, C Statistical distributions of particulate matter and error associated with sampling frequency. Atmospheric Environment 35: Seifeld, J.H. and Pandis, S.N., Atmospheric chemistry and physics: From air pollution to climate change. New York: Wiley. Singh, V.P., Gou, H. and Yu, F.X., Parameter estimation for 3 parameter log logistic distribution (LLD3) by Pome. Stochastic Hydrology and Hydraulics 7: Smith, L.R Extreme value analysis of environmental time series: An application to trend detection in ground level ozone. Statistical Science 44: Tiku, M.L. and Suresh, R.P A new method of estimation for location and scale parameters. Journal of Statistical Planning and Inference 30: Wingo, D.R. (Metrica) Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored Life Data. 40: doi: /bf Zaharim, A., Najid, S.K., Razali, A.M. and Sopian, K Analysing Malaysian wind speed data using statistical distribution. Proceedings of the 4th IASME/WSEAS International Conference on Energy and Environment, Cambridge, UK. 103 Journal of Energy in Southern Africa Vol 28 No 1 February 2017